# Sensitivity analysis of free torsional vibration frequencies of thin-walled laminated beams under axial load

## Abstract

The paper addresses sensitivity analysis of free torsional vibration frequencies of thin-walled beams of bisymmetric open cross section made of unidirectional fibre-reinforced laminate. The warping effect and the axial end load are taken into account. The consideration is based upon the classical theory of thin-walled beams of non-deformable cross section. The first-order sensitivity variation of the frequencies is derived with respect to the design variable variations. The beam cross-sectional dimensions and the material properties are assumed the design variables undergoing variations. The paper includes a numerical example related to simply supported I-beams and the distributions of sensitivity functions of frequencies along the beam axis. Accuracy is discussed of the first-order sensitivity analysis in the assessment of frequency changes due to the fibre volume fraction variable variations, and the effect of axial loads is discussed too.

## Keywords

Sensitivity analysis Free torsional vibration frequencies Thin-walled beams Analytical solution FEM## 1 Introduction

The present-day, thin-walled structures are frequently made of composite materials. The main reason is that they are becoming progressively stronger, lighter or less expensive, compared to traditional materials such as steel or aluminium. Hence, a composite material made of a polymer matrix reinforced with fibres (FRP) may be applied in the branches of civil engineering, ship structures, transportation industry and more.

The fibres are usually glass, carbon, aramid or basalt, and other fibre materials, e.g. paper, wood or asbestos are also applied. The polymer matrix is usually epoxy, vinyl ester or polyester thermosetting plastic. The orientation of reinforcing fibres affects strength and resistance to deformation of the polymer (the properties of composites). Unidirectional, bidirectional or random categories of composites, with respect to fibre alignment, are possible. Unidirectional composites show the greatest strength of all composites, while load is aligned with the fibres; in other directions, their strength decreases considerably depending on the matrix material. In the case of bidirectional composites, ultimate strength is lower than in the case of unidirectional composites, but if occurs in two directions; thus, the properties are more uniform in all loading directions. In the case of the random distribution of fibres, the parameters depend on fibre arrangement; in this case, material has usually the lowest strength. The strength of the material does not only depend on the orientation of the fibres but also on density of fibres in the matrix.

The application of unidirectional fibre-reinforced laminate structures in engineering increases due to their mechanical and economic advantages [4, 5, 12, 20]. In structural design, the constraints related to frequencies of free vibrations can be taken into account. If the constraints on frequency are not fulfilled, re-analysis is necessary to redirect the value in the admissible region. Here, dynamic re-analysis of structures with updated design variables may be replaced by sensitivity analysis of frequencies. Moreover, the results of sensitivity analysis lead to the advantageous region the design variable change, determining its necessary variations. The paper deals with the first-order sensitivity analysis of free torsional vibration frequencies of thin-walled beams of bisymmetric cross section made of unidirectional fibre-reinforced laminate subjected to axial end loads. The design variables undergoing variations are cross-sectional dimensions, (excluding cross-sectional height) and the properties of a laminate material. The beam behaviour is described according to the classical thin-walled members of non-deformable cross section [21]. Homogenization modelling of a laminate material [19] is based on the theory of mixtures cells. The first variation of free torsional vibration frequencies with respect to variations of the design variation is derived by means of variational calculus [6, 7, 13]. A numerical example of the paper deals with a simply supported I-beam. Here, fibre volume fraction is assumed the design variable under investigation. Sensitivity analysis of accuracy is discussed and compared with re-analysis results of the beam with updated parameters. The paper continues the prior research included in the paper [15].

## 2 First variation of free torsional vibration

Consider free torsional vibrations of an axially loaded thin-walled I-beam of bisymmetric cross section made of unidirectional fibre-reinforced laminate presented in Fig. 1. It is well known that the flexural and torsional vibrations for these kinds of cross sections are independent [21]. The torsional vibrations are described according to the classical theory of thin-walled beams of non-deformable cross section [21]. The analysis is focused on the single natural frequencies. The Cartesian coordinate system shown in Fig. 1 defines the *z*-axis representing the beam longitudinal axis and *x*- and *y*-axes representing cross-sectional symmetry axes.

*G*—homogenized shear modulus,

*v*—Poisson’s ratio in longitudinal direction, \(v_\mathrm{m}\), \(v_\mathrm{f}\)—Poisson’s ratios of matrix and fibres, and

*f*—fibre volume fraction,

*m*—homogenized mass density, \(m_\mathrm{m}\), \(m_\mathrm{f}\)—mass density of matrix and fibre.

*L*subjected to the axial end loads

*P*covers the warping stress energy (first term), the free torsion energy (second term) and the potential energy of the end axial loads (last term) [17]:

*A*—cross section area, (...)\('\)=d(...)/dz and (...)\(''\)—first and second derivatives with respect to axial coordinate

*z*.

*T*of a homogeneous beam of mass density

*m*is

It should be noted that the differential equation (8) is valid for a beam of bisymmetric variable cross section excluding its height.

*s*. The dimensions of the cross section and the material properties are assumed design variables. In order to derive the first variation of the square of the natural frequency with respect to the variation \(\mathrm{d}s\), the first variation of the functional (7) is computed

*s*. The latter integral in Eq. (10) expresses the virtual work of the internal forces on arbitrary virtual displacements, due to the virtual work theorem it equals zero. The last two terms in the first integral can be rewritten as

*F*(

*z*) represents variation of square frequency of torsional vibrations due to a variation of the design variable \(\delta s(z)\) in the cross section

*z*. Distribution of the function

*F*(

*z*) can be stated analytically or numerically by means of the finite element method.

Material properties for matrix, fibres and composite for \(f=5\), 50 and 95%

Matrix (Polyester) | Fibres (Boron) | Composite for \(f=5\)% | Composite for \(f=50\)% | Composite for \(f=95\)% | ||||
---|---|---|---|---|---|---|---|---|

LD | TD | LD | TD | LD | TD | |||

Mass density (kg/\(\hbox {m}^3\)) | \(m_\mathrm{m}=1380\) | \(m_\mathrm{f}=2450\) | \(m=1433\) | \(m=1915\) | \(m=2396\) | |||

Young’s modulus (GPa) | \(E_\mathrm{m}=2.50\) | \(E_\mathrm{f}=420\) | \(E_\mathrm{l}=23.34\) | \(E_\mathrm{t}=3.20\) | \(E_\mathrm{l}=211.25\) | \(E_\mathrm{t}=8.36\) | \(E_\mathrm{l}=399.12\) | \(E_\mathrm{t}=79.94\) |

\(D_\mathrm{l}=23.67\) | \(D_\mathrm{l}=212.01\) | \(D_\mathrm{l}=403.71\) | ||||||

Kirchhoff’s modulus (GPa) | \(G_\mathrm{m}=1.20\) | \(G_\mathrm{f}=170\) | \(\hbox {G}=1.28\) | \(G=3.20\) | \(G=36.35\) | |||

Poisson’s ratio (-) | \(v_\mathrm{m}=0.33\) | \(v_\mathrm{f}=0.20\) | \(v=0.30\) | \(v=0.24\) | \(v=0.20\) |

Geometrical characteristics of the I-section (\(L=3\) m, \(b=0.1\) m, \(h=0.2\) m, \(t_0=0.003\) m) (Fig. 1)

| \((2b+h)t_0\) | 0.0012 |

\(r_0\) (m) | \(\sqrt{\frac{J_0}{A}}=\frac{1}{2}\sqrt{\frac{2b^3+6bh^2+h^3}{6b+3h}}\) | 0.0842 |

\(J_{\omega }\) (\(\hbox {m}^6\)) | \(\frac{1}{24}b^3h^2t_0\) | 5.0 10\(^{-9}\) |

\(J_{d}\) (\(\hbox {m}^4\)) | \(\frac{1}{3}(2b+h)t_0^3\) | 3.6 10\(^{-9}\) |

\(J_0\) (\(\hbox {m}^4\)) | \(\frac{1}{12}(2b^3+6bh^2+h^3)\) | 8.5 10\(^{-6}\) |

## 3 Numerical examples

Square free torsional vibration frequency \(\omega ^2\) for \(n=1\) and \(f=5\), 50, 95% (\(L=3\) m, \(b=0.1\) m, \(h=0.2\) m, \(t_0=0.003\) m) (Fig. 1)

\(f=5\)% (\(P_{cr}^{FEM}=6\, \)kN) | \(f=50\)% (\(P_{cr}^{FEM}=22\) kN) | \(f=95\%\) (\(P_{cr}^{FEM}=139\, \hbox {kN}\)) | |||||||
---|---|---|---|---|---|---|---|---|---|

Analytical approach (15) (rad/s)\(^2\) | FEM (rad/s)\(^2\) | Diff.% | Analytical approach (15) (rad/s)\(^2\) | FEM (rad/s)\(^2\) | Diff.% | Analytical approach (15) (rad/s)\(^2\) | FEM (rad/s)\(^2\) | Diff.% | |

\(P=5\, \hbox {kN}\) (Compressive load) | 8900.1 | 8423.3 | 5.4 | 76,552.0 | 70,395.0 | 8.0 | 123,856.0 | 120,319.0 | 2.9 |

\(P=0\, \hbox {kN}\) | 12,085.5 | 11,590.0 | 4.1 | 78,936.5\(^{\hbox {a}}\) | 72,765.5 | 7.8 | 125,762.0 | 122,214.0 | 2.8 |

\(P=-5 \,\hbox {kN}\) (Tensile load) | 15,270.9 | 14,756.0 | 3.4 | 81,321.0 | 75,135.0 | 7.6 | 127,667.0 | 124,108.0 | 2.8 |

*P*due to different load values: \(P=-5 \,\hbox {kN}\) (tension) or 5 kN (compression), in the case of the analytical solution ranging from \(-50\) to 50 kN.

The results of analytical and numerical analyses are shown in Table 3 and in Figs. 3, 4, 5, 6 and 7. Table 3 compares two solutions of square free vibration frequency, i.e. the analytical solution (15) proposed in the paper with the FEM solution for three different axial loads. Figures 3, 4 and 5 show sensitivity functions of square free vibration frequency (16) with regard to fibre volume fraction parameter f along the I-beam axis in the case of axial tensile and compressive end loads and without a load, considering two different numbers of half-waves \(n=1\) and 2. Furthermore, Fig. 6 shows relative values of sensitivity functions of square free torsional vibration frequency, related to the axial end load *P* regarding different fibre volume fraction parameters a) \(f=5\%\), b) \(f=50\%\), c) \(f=95\%\) along the I-beam axis due to a number of half-waves \(n=1\). Figure 7 show the main solutions, i.e. first-order sensitivity analysis compared with the FEM based due to a relative values of square free torsional vibration frequency function of the I-beam under axial loads depending on the fibre volume fraction parameters, regarding number of the half-waves \(n=1\).

The numerical analysis carried out indicates high convergence of the results of analytical and numerical approaches. The differences between solutions remain at an average level of 10%.

the square free torsional vibration frequency increases with the fibre volume rise in a composite material; it decreases while the fibre volume fraction in the material is reduced (see Table 3),

the difference between analytical and numerical solutions is directly proportional to the homogeneity extent of the composite material, i.e. in the case of a highly homogeneous material the differences are smaller; oppositely, while material is more heterogeneous the differences are greater (Table 3),

the influence of compressive or tensile forces on the square free torsional vibration frequency is negligible, as shown in the results presented in Figs. 3 and 7,

the numerical analysis indicates that square free torsional vibration frequency with regard to fibre volume is weakly nonlinear; thus, the first-order sensitivity analysis (linear solution) is a suitable approximation (with an accuracy of 10–15%) of the solution, ranging from 20 to 80% of the fibre volume fraction (Fig. 7).

## 4 Conclusions

The paper deals with the first-order sensitivity analysis of the square free torsional vibration with regard to the fibre volume. Analytical solution of the problem is investigated and compared with the FEM solution. The first-order sensitivity analysis proves a sufficient approximation of the numerical FEM solution. The proposed analytical solution based on classical theory of thin-walled beams of non-deformable cross sections, taking into account the warping effect, showed its validity in the analysis of such a kind of structures. The differences between compared solutions, the FEM solution and a linear approach to the sensitivity analysis are acceptable from an engineering point of view, remaining at an average level of 10%. Finally, it should be emphasized that the proposed simplified solution based on the sensitivity analysis seems a useful tool in the optimal design or the analysis of beams with variable cross sections and mechanical properties of the beam material.

## Notes

### Acknowledgements

The calculations presented in the paper were carried out at the TASK Academic Computer Centre in Gdańsk, Poland.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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